my %id2seq = (); # this is the verbose form of my %id2seq;
my $id = '';     # this is the wrong place to declare this var! declare it when you need it ie
                 # inside the while(<File>){ block
         # missing the mode: put always even if it defaults to '<'
open File,"human_hg19_circRNAs_putative_spliced_sequence.fa",or die $!;
         # better use lexical filehandle like in open my $fh, '<', $filepath or die
         # bareword is still accepted but by onvention is UPPERCASE so no open File...

    while(<File>){
        chomp;
                 # here you are capturing something: if you want just the part before | you
                 # have here the possibility to get it:  /^>([\w\d]+\|)/ as starting option?
        if($_ =~ /^>(.+)/){
                 # or here:
                 $id = $1;
                 # cutting $1 like in: $id = (split /\|/, $1)[0]
...
           # AHHH! this is error! are your use strict; use warnings just make-up?
           # it must be foreach my $id ..
           # (or really does not raise a warn for the scope you given to $id ??? if so is even
           #  worst!!)
           # in short: pay attention to the scope of your variables
   foreach $id (keys %id2seq){
                 # here the last good possibility to cut $id:
                 # $id = (split /\|/, $id)[0];
        if (-f $id){                             # this is a lie..
            print $id." Already exists, about to override it","\n"
        }                 # .. because you are going to append, not to overwrite
        open  my $out_fh, '>>', "$id.fa" or die $!; 
                     # here parens are unneeded and probably nasty
        print $out_fh (">".$id."\n",$id2seq{$id}, "\n");